On the meaning of the word "generic" in Lie Algebra (or otherwise)

Question

I always have a problem with the word generic in the literature of mathematics. Let me ask you a specific question about "non-degenerate $\mathbb{Z}$-graded lie algebras''. The definition I'm working with says:

A $\mathbb{Z}$-graded lie algebra (over $\mathbb{C}$) $\mathfrak{g}$, i.e. $\mathfrak{g}=\bigoplus_{n\in \mathbb{Z}}\mathfrak{g}_n$, is non-degenerate if the following are satisfied:

• $\mathfrak{g}_n$ are finite dimensional for all $n\in \mathbb{Z}$.
• $\mathfrak{g}_0$ is abelian.
• For any $n\in \mathbb{Z}_{>0}$, and generic $\lambda\in \mathfrak{g}_0^*$, the pairing $\mathfrak{g}_n\times \mathfrak{g}_{-n}\to \mathbb{C}$ given by $(x,y)\mapsto \lambda([xy])$ is non-degenerate.

Everything in this definition is perfectly clear to me except this word "generic". What is meant by a generic dual vector? I would ask what is meant by generic in general, but probably that's different according to the context.

Answer 1

As you say, it depends on context. Vaguely, it means something like "every element except for the elements in a 'small' subset," where the meaning of 'small' depends on context. In this context it might mean either

• every element except for the elements in a finite (or maybe countably infinite) union of affine subspaces, or
• every element except for the elements in a finite (or maybe countably infinite) union of Zariski closed subvarieties.

I don't know which the author intends.

Answer 2

Here is the heuristic I use: roughly, there are two notions of "generic" in mathematics, each dual to the other. If you're working in some structure $M$ with a distinguished collection of substructures $N_i \hookrightarrow M$, one can either say that an $m \in M$ is generic if it lies outside of all the $N_i$, or one can say that an $m \in M$ is generic if it lies in all of the $N_i$.

For example, if you're working in a topological space, one notion of a "generic point" could be one that lies in every closed subset. This corresponds to the latter notion.

For another example, take a definable set of finite Morley rank $n$ in a first-order structure. A generic point of this set is one that lies in no definable subset of rank less than $n$. This corresponds to the former notion.